Complex Analysis - Maths KU

S
y
(a) A circular sector
3π/8
v
3
3π/4
w = z 1/2
S′
π/4
√3
(b) The image of the set in (a)
Figure 2.30 The mapping w = z 1/2
x
u
2.4 Special Power Functions 93
principal argument. That is, if w = z1/2 , then we have |w| = � |z| and
Arg(w) = 1
2Arg(z). These relations follow directlyfrom (7) and are helpful
in determining the images of sets under w = z1/2 .
EXAMPLE 8 Image of a Circular Sector under w = z 1/2
Find the image of the set S defined by |z| ≤3, π/2 ≤ arg(z) ≤ 3π/4, under
the principal square root function.
Solution Let S ′ denote the image of S under w = z 1/2 . Since |z| ≤ 3
for points in S and since z 1/2 takes the square root of the modulus of a
point, we must have that |w| ≤ √ 3 for points w in S ′ . In addition, since
π/2 ≤ arg(z) ≤ 3π/4 for points in S and since z 1/2 halves the argument of a
point, it follows that π/4 ≤ arg(w) ≤ 3π/8 for points w in S ′ . Therefore, we
have shown that the set S shown in color in Figure 2.30(a) is mapped onto
the set S ′ shown in grayin Figure 2.30(b) byw = z 1/2 .
Principal nth Root Function Bymodifying the argument given
on page 91 that the function f(z) = z 2 is one-to-one on the set defined
by −π/2 < arg(z) ≤ π/2, we can show that the complex power function
f(z) =z n , n>2, is one-to-one on the set defined by
− π
n
π
< arg(z) ≤ . (13)
n
It is also relativelyeasyto see that the image of the set defined by(13) under
the mapping w = z n is the entire complex plane C excluding w = 0. Therefore,
there is a well-defined inverse function for f. Analogous to the case n =2,
this inverse function of z n is called the principal nth root function z 1/n .
The domain of z 1/n is the set of all nonzero complex numbers, and the range
of z 1/n is the set of complex numbers w satisfying −π/n < arg(w) ≤ π/n. A
purelyalgebraic description of the principal nth root function is given bythe
following formula, which is analogous to (7).
Definition 2.6 Principal nth Root Functions
For n ≥ 2, the function z 1/n defined by:
z 1/n = n� |z|e iArg(z)/n
is called the principal nth root function.
(14)
Notice that the principal square root function z 1/2 from Definition 2.4
is simplya special case of (14) with n = 2. Notice also that in Definition
2.6 we use the symbol z 1/n to represent something different than the same
symbol used in Section 1.4. As with the symbol z 1/2 , whether z 1/n represents